[/quote]
,
such that![\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]](http://latex.artofproblemsolving.com/3/1/9/319c9a2a80c4da5961cd5389af0606fa429ca8e2.png)
for any positive integer 
.
be the incircle of triangle 
and 
at 
.
meets 
.
be 
-
be the midpoint of 
,
are concyclic.
Prove that
and incentre 
.
intersects the lines 
,
,
at points 
,
,
,
,
,
,
of the segments 
,
,
.
and 
be concentric circles, with 
to 
)
and 
on 
.
where 
.
is the circumcenter of triangle 
is the projection of point 
onto line 
.
,
are 
,
,
,
intersects 
and 
at points 
and 
,
.
are consecutive vertices of a regular 
-
and 
are tangents to the circle center 
.
is the intersection point of 
and 
.
are collinear.
and 
respectively. If 
intersect the line through 
,
.
and circumcircle 
.
,
again at 
.
at 
and 
intersect on 
with 
of 
onto 
intersects 
at 
and 
is an isosceles triangle.
be a quadrilateral inscribed in a circle with center 
be the circumcircle of 
and 
be the circumcircle of 
.
.
.
.
.
,
,
are positive integers 
.
is also an integer and equals 
.
,
such that 
and 
are perpendicular and parallel to bisector of 
and 
bisect two arcs 
.
Y by
In the plane, 
points are marked, no three of which are collinear. All possible segments between them are drawn. Grisha assigned to each drawn segment a real number with absolute value no greater than 
. For every group of 
marked points, he calculated the sum of the numbers on all 
connecting segments. It turned out that the absolute value of each such sum is at least 
, and there are both positive and negative such sums. What is the maximum possible value of ?
points are marked, no three of which are collinear. All possible segments between them are drawn. Grisha assigned to each drawn segment a real number with absolute value no greater than 
. For every group of 
marked points, he calculated the sum of the numbers on all 
connecting segments. It turned out that the absolute value of each such sum is at least 
, and there are both positive and negative such sums. What is the maximum possible value of ?This post has been edited 2 times. Last edited by egxa, Apr 21, 2025, 7:22 PM
,
be the sum corresponding to the subset of 
.
and 
.
be the sum over all segments 
with: 
,
,
By switching the roles of positive and negative, we also get 
we have 
.
or 
then 
.
then 
.
.
,
to all remaining segments. In this construction, the sum over any 
.
.
